Generalized Auto-calibrating Partially Parallel Acquisitions (GRAPPA) is described in Griswold, et al., Proceedings of the ISMRM, Vol. 7, Issue 6, Pg. 1202-1210 (2002). GRAPPA facilitates generating uncombined coil images for coils in an array of receive coils used by a parallel magnetic resonance imaging (pMRI) apparatus. GRAPPA reconstructs missing lines in coil elements by forming linear combinations of neighboring lines to reconstruct individual missing data points. The weights for these linear combinations are derived by forming a fit between additionally acquired lines using a pseudo-inverse operation.
GRAPPA Operator Gridding (GROG) is described in Seiberlich, et al., Non-Cartesian data reconstruction using GRAPPA operator gridding (GROG), Magn Reson Med. 2007 December; 58(6): 1257-65. GROG facilitates gridding data sampled along a non-Cartesian trajectory. GROG facilitates shifting acquired data points to another (e.g., nearest) Cartesian location to facilitate converting non-Cartesian to Cartesian data. GROG synthesizes the net weight for a shift from a basis set of weights along logical k-space directions. GROG employs local averaging because the reconstructed points fall upon the Cartesian grid. This facilitates not having to calculate and apply a density compensation function (DCF).
Magnetic resonance imaging (MRI) pulse sequences manipulate gradient fields by controlling gradient coils. Gradient coils are physical things that have physical properties including, for example, a delay time and a slew rate. The delay time describes how quickly a gradient coil may respond to a direction to change the state of the gradient coil. The slew rate describes the rate of ascent or descent of a gradient from zero to its maximum amplitude once it has begun to respond to the direction to change its state. Having a faster slew rate allows the gradient to slew from zero to its maximum amplitude in less time, which in turn facilitates having faster gradients and shorter echo spacing. Unfortunately, different gradient coils may have different delay times and may have different slew rates, which may introduce artifacts into magnetic resonance images.
FIG. 1 illustrates a square wave 100 and a non-square wave 110. Square wave 100 and non-square wave 110 represent the amplitude of a gradient field being produced by a gradient coil. In theory a gradient coil would respond instantaneously to produce a gradient field whose amplitude would then transition like square wave 100. In practice, due to slew rate, a gradient coil responds less than instantaneously and produces a gradient field whose amplitude transitions more like wave 110.
FIG. 2 illustrates a non-square wave 200 that represents the amplitude of a gradient field produced by a gradient coil. Non-square wave 200 illustrates the effect of both slew rate and delay. For example, a gradient coil may receive an input at time T20 that is intended to cause the gradient coil to change its state. However, the gradient coil may not begin to slew until time T21. The delay may be caused, for example, by switching, by a capacitor charging, or by other factors. Once the gradient coil begins to slew at T21, the gradient field amplitude continues to change until time T22 where the gradient field amplitude achieves steady state. At a later time T23 the gradient coil may slew in the other direction until the gradient field returns to its original state at time T24.
MRI pulse sequences may manipulate multiple gradients at the same time. Thus, the situation illustrated in FIG. 3 could occur. FIG. 3 illustrates the amplitude of a gradient field 300 transitioning as controlled by a gradient coil GX and the amplitude of a gradient field 310 transitioning as controlled by a gradient coil GY. While gradient coils GX and GY have similar slew rates, they have different delay times.
Both GX and GY may be controlled to change their state at time T30. GX may begin to respond at time T31 while GY does not begin to respond until time T32. GX achieves steady state at time T32 while GY does not achieve steady state until T33. Between T33 and T35, both gradient fields are in steady state. A trajectory associated with the two gradient fields GX and GY may be stable during this period of time. Both GX and GY may be controlled at time T34 to change their state. Once again GX may respond more quickly and begin to change state at T35 while GY does not respond until a later time. Eventually both GX and GY achieve their original steady state.
During the period of time 36 between T31 and T32, only the GX gradient field is active and that field has not yet achieved steady state. During the period of time 37 between T32 and T33, both the GX and the GY gradients are active but the GY field has not yet achieved steady state. Thus, a trajectory associated with GX and GY may not be stable during the period of time between T31 and T33. Once GX and GY achieve steady state at T33 the trajectory may be stable during, for example, period of time 38. The situation illustrated in FIG. 3 could be even further complicated if GX and GY also had different slew rates.
FIG. 4 illustrates a radial projection associated with a system where gradient coils reacted identically and without delay. The radial projection includes a portion 400OUT that extends directly out from the center of k-space. The radial projection also includes a portion 400BACK that returns directly through the center of k-space. Note that the projection angle θ is the same for both 400OUT and 400BACK.
FIG. 5 also illustrates a radial projection. But in FIG. 5 the radial projection is associated with a system where gradient coils did not react identically and did not react instantaneously. The radial projection includes a portion 500OUT that extends out from a position offset from the center of k-space. The radial projection also includes a portion 500BACK that returns but not directly through the center of k-space. Once again note that the projection angle is the same for both 500OUT and 500BACK.
FIG. 6 illustrates a trajectory that would be experienced by the radial projection that includes 500OUT and 500BACK. Once the gradients reach steady state, the projection is stable and lies along the desired projection angle. However, the projection does not originate from the center of k-space nor does it pass back through the center of k-space. This may lead to artifacts in an image reconstructed from the radial projection.
FIGS. 5 and 6 illustrate how gradient delays and stewing can cause a trajectory desired by a pulse sequence to not be exactly where it was intended to be. The trajectory may proceed at the correct angle during an “on angle” portion, but it may not pass through desired points (e.g., center of k-space). Ideally, a projection would go out and back passing through the same points. In FIG. 6, since the X gradient reacts more quickly than the Y gradient, the projection first gets shifted to the right in the X direction and then gets shifted to the left in the X direction. Although neither 500OUT nor 500BACK are exactly where they are supposed to be, they are still useful because they are on the desired projection angle θ.
To summarize, gradient timing delays may cause mismatches between a desired trajectory and an actual trajectory. In radial scanning, individual projections may be shifted along the direction of the projection or may be translated in k-space so that they do not pass through the center of k-space. These shifts may cause artifacts in reconstructed images. Conventional systems may attempt to address the shifts by measuring the trajectory using a separate acquisition. Making a separate acquisition takes additional time, during which conditions may change. These conventional approaches assume that shifts are consistent between measurements. However, the assumption may not hold due to gradient coupling, patient motion, or other factors. Even if the assumption holds during the additional acquisition, the conventional approaches may still only provide a partial solution. The extra measurement may not address k-space signals that are being used for additional purposes including, for example, self-gating signals acquired from repeatedly sampling the echo peak.